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### multiplying radical expressions with variables

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18The factors $$(a+b)$$ and $$(a-b)$$ are conjugates. Multiply: $$3 \sqrt { 6 } \cdot 5 \sqrt { 2 }$$. To multiply ... Access these online resources for additional instruction and practice with adding, subtracting, and multiplying radical expressions. As with multiplication, the main idea here is that sometimes it makes sense to divide and then simplify, and other times it makes sense to simplify and then divide. \\ & = \frac { x - 2 \sqrt { x y } + y } { x - y } \end{aligned}\), $$\frac { x - 2 \sqrt { x y } + y } { x - y }$$, Rationalize the denominator: $$\frac { 2 \sqrt { 3 } } { 5 - \sqrt { 3 } }$$, Multiply. \\ & = \frac { \sqrt { 25 x ^ { 3 } y ^ { 3 } } } { \sqrt { 4 } } \\ & = \frac { 5 x y \sqrt { x y } } { 2 } \end{aligned}\). $\begin{array}{c}\frac{\sqrt{16\cdot 3}}{\sqrt{25}}\\\\\text{or}\\\\\frac{\sqrt{4\cdot 4\cdot 3}}{\sqrt{5\cdot 5}}\end{array}$, $\begin{array}{r}\frac{\sqrt{{{(4)}^{2}}\cdot 3}}{\sqrt{{{(5)}^{2}}}}\\\\\frac{\sqrt{{{(4)}^{2}}}\cdot \sqrt{3}}{\sqrt{{{(5)}^{2}}}}\end{array}$, $\frac{4\cdot \sqrt{3}}{5}$. $\begin{array}{l}\sqrt{\frac{8\cdot 3\cdot x\cdot {{y}^{3}}\cdot y}{8\cdot y}}\\\\\sqrt{\frac{3\cdot x\cdot {{y}^{3}}}{1}\cdot \frac{8y}{8y}}\\\\\sqrt{\frac{3\cdot x\cdot {{y}^{3}}}{1}\cdot 1}\end{array}$. $$\begin{array} { c } { \color{Cerulean} { Radical\:expression\quad Rational\: denominator } } \\ { \frac { 1 } { \sqrt { 2 } } \quad\quad\quad=\quad\quad\quad\quad \frac { \sqrt { 2 } } { 2 } } \end{array}$$. Simplify. This next example is slightly more complicated because there are more than two radicals being multiplied. Notice that the process for dividing these is the same as it is for dividing integers. Give the exact answer and the approximate answer rounded to the nearest hundredth. We just have to work with variables as well as numbers. }\\ & = \frac { 3 \sqrt [ 3 ] { 4 a b } } { 2 b } \end{aligned}\), $$\frac { 3 \sqrt [ 3 ] { 4 a b } } { 2 b }$$, Rationalize the denominator: $$\frac { 2 x \sqrt [ 5 ] { 5 } } { \sqrt [ 5 ] { 4 x ^ { 3 } y } }$$, In this example, we will multiply by $$1$$ in the form $$\frac { \sqrt [ 5 ] { 2 ^ { 3 } x ^ { 2 } y ^ { 4 } } } { \sqrt [ 5 ] { 2 ^ { 3 } x ^ { 2 } y ^ { 4 } } }$$, \begin{aligned} \frac{2x\sqrt{5}}{\sqrt{4x^{3}y}} & = \frac{2x\sqrt{5}}{\sqrt{2^{2}x^{3}y}}\cdot\color{Cerulean}{\frac{\sqrt{2^{3}x^{2}y^{4}}}{\sqrt{2^{3}x^{2}y^{4}}} \:\:Multiply\:by\:the\:fifth\:root\:of\:factors\:that\:result\:in\:pairs.} and ; Spec You multiply radical expressions that contain variables in the same manner. Simplify, using $\sqrt{{{x}^{2}}}=\left| x \right|$. ), Rationalize the denominator. Divide: \(\frac { \sqrt [ 3 ] { 96 } } { \sqrt [ 3 ] { 6 } }. With some practice, you may be able to tell which is easier before you approach the problem, but either order will work for all problems. The Quotient Raised to a Power Rule states that ${{\left( \frac{a}{b} \right)}^{x}}=\frac{{{a}^{x}}}{{{b}^{x}}}$. A radical is an expression or a number under the root symbol. The radicand in the denominator determines the factors that you need to use to rationalize it. $\frac{\sqrt{48}}{\sqrt{25}}$. In our last video, we show more examples of simplifying radicals that contain quotients with variables. Rationalize the denominator: $$\frac { 3 a \sqrt { 2 } } { \sqrt { 6 a b } }$$. In the next example, we will use the same product from above to show that you can simplify before multiplying and get the same result. Rationalize the denominator: $$\sqrt [ 3 ] { \frac { 27 a } { 2 b ^ { 2 } } }$$. A radical is a number or an expression under the root symbol. \\ &= \frac { \sqrt { 20 } - \sqrt { 60 } } { 2 - 6 } \quad\quad\quad\quad\quad\quad\:\:\:\color{Cerulean}{Simplify.} Well, what if you are dealing with a quotient instead of a product? This algebra video tutorial explains how to divide radical expressions with variables and exponents. (Assume $$y$$ is positive.). The answer is $y\,\sqrt{3x}$. Notice this expression is multiplying three radicals with the same (fourth) root. \\ & = \frac { 3 \sqrt [ 3 ] { a } } { \sqrt [ 3 ] { 2 b ^ { 2 } } } \cdot \color{Cerulean}{\frac { \sqrt [ 3 ] { 2 ^ { 2 } b } } { \sqrt [ 3 ] { 2 ^ { 2 } b } }\:\:\:Multiply\:by\:the\:cube\:root\:of\:factors\:that\:result\:in\:powers.} In the following video, we show more examples of multiplying cube roots. Learn more Accept. \begin{aligned} \sqrt [ 3 ] { \frac { 27 a } { 2 b ^ { 2 } } } & = \frac { \sqrt [ 3 ] { 3 ^ { 3 } a } } { \sqrt [ 3 ] { 2 b ^ { 2 } } } \quad\quad\quad\quad\color{Cerulean}{Apply\:the\:quotient\:rule\:for\:radicals.} \\ ( \sqrt { x } + \sqrt { y } ) ( \sqrt { x } - \sqrt { y } ) & = ( \sqrt { x } ) ^ { 2 } - ( \sqrt { y } ) ^ { 2 } \\ & = x - y \end{aligned}, Multiply: $$( 3 - 2 \sqrt { y } ) ( 3 + 2 \sqrt { y } )$$. The basic steps follow. \\ &= \frac { \sqrt { 4 \cdot 5 } - \sqrt { 4 \cdot 15 } } { - 4 } \\ &= \frac { 2 \sqrt { 5 } - 2 \sqrt { 15 } } { - 4 } \\ &=\frac{2(\sqrt{5}-\sqrt{15})}{-4} \\ &= \frac { \sqrt { 5 } - \sqrt { 15 } } { - 2 } = - \frac { \sqrt { 5 } - \sqrt { 15 } } { 2 } = \frac { - \sqrt { 5 } + \sqrt { 15 } } { 2 } \end{aligned}\), $$\frac { \sqrt { 15 } - \sqrt { 5 } } { 2 }$$. Apply the distributive property when multiplying a radical expression with multiple terms. Assume \ ( \frac { \sqrt { x } } { \sqrt { { { { x. 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